Material Property Estimation Using Inverse Interpolation

ABSTRACT

Magnetic field sensor probes are disclosed which comprise primary or drive windings having a plurality of current carrying segments. The relative magnitude and direction of current in each segment are adjusted so that the resulting interrogating magnetic field follows a desired spatial distribution. By changing the current in each segment, more than one spatial distribution for the magnetic field can be imposed within the same sensor footprint. Example envelopes for the current distributions approximate a sinusoid in Cartesian coordinates or a first-order Bessel function in polar coordinates. One or more sensing elements are used to determine the response of a test material to the magnetic field. These sense elements can be configured into linear or circumferential arrays.

RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.12/291,308, filed Nov. 7, 2008, which application is a divisional ofU.S. application Ser. No. 11/227,307, filed Sep. 15, 2005, now U.S. Pat.No. 7,467,057, which is a divisional of U.S. application Ser. No.10/045,650, filed Nov. 8, 2001, now U.S. Pat. No. 6,992,482, whichclaims the benefit of U.S. Provisional Application No. 60/246,853, filedNov. 8, 2000, U.S. Provisional Application No. 60/275,754 filed Mar. 14,2001, U.S. Provisional Application No. 60/277,532 filed Mar. 21, 2001,and U.S. Provisional Application No. 60/284,972 filed Apr. 19, 2001.

The entire teachings of the above application(s) are incorporated hereinby reference.

BACKGROUND

The technical field of this invention is that of nondestructivematerials characterization, particularly quantitative, model-basedcharacterization of surface, near-surface, and bulk material conditionfor flat and curved parts or components using eddy-current sensors.Characterization of bulk material condition includes (1) measurement ofchanges in material state caused by fatigue damage, creep damage,thermal exposure, or plastic deformation; (2) assessment of residualstresses and applied loads; and (3) assessment of processing-relatedconditions, for example from shot peening, roll burnishing,thermal-spray coating, or heat treatment. It also includes measurementscharacterizing material, such as alloy type, and material states, suchas porosity and temperature. Characterization of surface andnear-surface conditions includes measurements of surface roughness,displacement or changes in relative position, coating thickness, andcoating condition. Each of these also includes detection ofelectromagnetic property changes associated with single or multiplecracks. Spatially periodic field eddy-current sensors have been used tomeasure foil thickness, characterize coatings, and measure porosity, aswell as to measure property profiles as a function of depth into a part,as disclosed in U.S. Pat. Nos. 5,015,951 and 5,453,689.

Conventional eddy-current sensing involves the excitation of aconducting winding, the primary, with an electric current source ofprescribed frequency. This produces a time-varying magnetic field at thesame frequency, which in turn is detected with a sensing winding, thesecondary. The spatial distribution of the magnetic field and the fieldmeasured by the secondary is influenced by the proximity and physicalproperties (electrical conductivity and magnetic permeability) of nearbymaterials. When the sensor is intentionally placed in close proximity toa test material, the physical properties of the material can be deducedfrom measurements of the impedance between the primary and secondarywindings. Traditionally, scanning of eddy-current sensors across thematerial surface is then used to detect flaws, such as cracks.

One of the difficulties encountered when performing material propertymeasurements with traditional spatially periodic field and other eddycurrent sensors is the limited sensitivity to flaws or defects, such ascracks, voids, inclusion, and corrosion, hidden behind metal layers ordeep within the test material. A limiting factor for these measurementsis often the frequency range of operation for the sensing device as itaffects both the depth of penetration of the magnetic field into thetest material and the detectable signal level of the device. The depthof penetration of the magnetic field into a material is determined bythe geometry of the drive winding and the skin depth for the magneticfield in the material. The geometry affects the dominant spatialwavelength for the decay of the field into the material. For measurementsensitivity to a hidden flaw, both the dominant spatial wavelength andthe skin depth need to be comparable to, or larger than, the thicknessof material between the flaw and the sensor. Since the skin depth variesinversely with the square root of the frequency, sensitivity to deepflaws requires low excitation frequencies. The excitation frequency alsoaffects the output signal level, which is the induced voltage from asecondary coil or winding for a traditional eddy current sensor. Thisvoltage is proportional to the rate of change of the magnetic fluxthrough the coil, and hence the excitation frequency. Since the inducedvoltage decreases with frequency, the lowest detectable signal leveldetermines the lowest frequency of operation for the sensor, which maynot be low enough for the detection of a flaw, according to the skindepth.

In addition, new materials, manufacturing processes and structuraldesigns, as well as new damage mechanisms, pose continual challenges tothe state-of-the-art in non-destructive evaluation. In particular, thicksections and multi-layered structures create difficult to inspect areasin which corrosion or other damage can propagate undetected. AlthoughX-ray and ultrasonics have become common for inspection of thickstructures and components (lapjoints, friction stir welds, turbineengine disks, and structural castings) for defects and geometricfeatures, these techniques are limited in their sensitivity and imageresolution. More importantly, they provide little, if any, informationon absolute material properties. There is a need for lower cost, higherspeed, wide area scanning capabilities not only to image defects, hiddencorrosion, and geometric features, but also to provide images ofmetallurgical properties and residual stresses (e.g., for ferrous alloysmagnetic permeability varies directly with applied and residual stress).

Giant magnetoresistive (GMR) and magnetoresistive sensing elements havebeen used to address this issue. Goldfine et al., described the use ofarrays of magnetoresistive sensors with meandering drive windings inU.S. Pat. No. 5,793,206 as an alternative to inductive coils. Wincheskiet. al. at NASA (Wincheski, 2001) and Raymond Rempt of Boeing (Rempt,2001) have used single sensing elements and arrays of GMR ormagnetoresistive sensing elements to detect subsurface cracks orcorrosion.

A GMR sensor offers substantial new capabilities at a very reasonablecost compared to competing technologies, such as SQUIDS. GMRs takeadvantage of the large magnetoresistive effect exhibited by certainmetallic magnetic superlattices. Whereas normal magnetoresistivematerials exhibit maximum changes in resistance on the order of 5% whenexposed to magnetic fields, GMRs exhibit resistance changes of 20% ormore. Giant magnetoresistance was first observed in Fe/Cr magneticsuperlattices, where a drop of as much as 45% of the resistivity wasmeasured at liquid helium temperature. At room temperature, themagnitude of the effect was reduced to about 12%. Other material systemshave been tested since then, with the Co/Cu magnetic superlatticeemerging as the system of choice in the development of practicalsensors. It exhibits resistivity drops of up to 55% at liquid heliumtemperature and 40% at room temperature.

A quantitative physical model of the giant magnetoresistive effect,developed by R. Q. Hood and L. M. Falicov, concludes that a largedifference in interface scattering for the different spins is needed toexplain the observed large GMR values. The magnetic superlattices havealternating layers of nonferromagnetic and ferromagnetic metals. Thethickness of the nonferromagnetic layers is chosen such that in theabsence of applied external magnetic field, the moments of consecutiveferromagnetic layers align antiparallel to each other. Thisantiferromagnetic coupling between these layers has been ascribed toindirect exchange interactions through the nonferromagnetic layers. Thepresence of an external field acts to align the moments of theferromagnetic layers, resulting in reduction of the electricresistivity.

The sensitivity of GMRs to magnetic field strength and direction, asopposed to the rate of change of magnetic field strength, suggests thefeasibility of a deep penetration eddy current type sensor. Typical eddycurrent devices lose sensitivity at lower frequencies. In order toachieve deep penetration of eddy currents, however, the excitationfrequency must be decreased in order to increase the skin depth. Thelack of sensitivity of simple inductive coils at low frequencies limitsthe depth of sensitivity of typical eddy current sensors in aluminum,for example, to a few millimeters. Replacing inductive coils or senseelements with GMR sensing elements has the potential to increase thedepth of sensitivity. The term depth of sensitivity, not penetration, isused because the sensing elements do not affect the magnetic field depthof penetration provided by the drive windings.

Some progress has been made in adapting GMR sensors to non-destructivetesting applications. Wincheski and Namkung have integrated a GMR with aself nulling probe driver coil to produce the Very Low Frequency (VLF)Self Nulling Probe. They have operated this device at excitationfrequencies down to 135 Hz and have used it to detect an EDM notch at adepth of up to 10 mm (Wincheski, 2001). However, the need still existsto improve measurement reliability and robustness with GMR andmagnetoresistive sensing element for eddy current and also for DCmeasurements with current driven drive windings.

SUMMARY

Aspects of the inventions described herein relates to methods andapparatus for the nondestructive measurements of materials using sensorsthat apply electromagnetic fields to a test material and detect changesin the electromagnetic fields due to the proximity and properties of thetest material. Novel drive winding patterns are described which promoteaccurate modeling, varying the depth of penetration into the testmaterial, and a reduced sensitivity to undesired and unmodeled parasiticresponse. The use of alternative sensors, such as GMRs, in the sensingelements is also described for improved sensitivity to low frequency,even dc, excitations. Arrays of these sensing elements promote imagingof surface and volumetric material property variations.

In one embodiment of this invention, alternative sensing elements areused with drive winding patterns that impose a magnetic fielddistribution on the test material. These sensing elements, such asmagnetoresistive, GMR, Hall Effect, and SQUIDs, permit lower frequencyoperation than inductive coils used in conventional eddy currentsensors. DC excitations can also be used. In a preferred embodiment, aGMR sensor is used as the sensing element and biased to operate in aregime where the output voltage of the GMR sensor is linear withmagnetic field intensity. In another preferred embodiment, the GMRsensor is encircled by a feedback coil that biases the sensor andmaintains a constant magnetic flux density in the vicinity of thesensor. The feedback current to this coil is performed by an electroniccircuit. This mode of operation provides a larger dynamic range than aGMR sensor by itself and its ability to measure magnetic fields is onlylimited by the amplitude of the current that the electronic circuit candrive through the feedback coil. In another embodiment, high frequencymeasurements outside the operating range of the feedback circuit can beperformed by operating the GMR sensor open-loop, without feedback, or bymeasuring the response of the feedback coil directly as with aconventional eddy current sensor.

When used with a shaped field distribution from the drive winding, theresponse of the GMR sensors can be accurately modeled. In oneembodiment, these models are used with measurements of the sensorresponse in air to calibrate the sensor so that absolute propertymeasurements can be obtained from the inspection of the test material.In another embodiment, these models can be used with measurements of thesensor response on a reference material, again so that absolute propertymeasurements can be obtained from the test material. Properties that canbe measured include the electrical conductivity, magnetic permeability,layer thicknesses and spatial property profiles, and the lift-off orproximity of the sensor to the test material. Measurements of thesematerial properties can also be correlated to other physical properties,such as residual and applied stresses in magnetizable materials ortemperature profiles across a material, and the presence of flaws ormaterial degradation, such as corrosion, crack initiation and growth,detection of inclusions, and porosity. In one embodiment, low frequencymeasurements of the magnetic permeability are combined with highfrequency measurements to determine the electrical conductivity.

In another embodiment, the model is used to create databases of responseand measurement grids so that table look-up algorithms can be used todetermine property values. These grids can be used, for example, todetermine the electrical conductivity and lift-off of a test material,the electrical conductivity and thickness of a material given thelift-off value, and the magnetic permeability of a test material andlift-off, given the thickness of the test material.

In an embodiment of this invention, the drive-winding geometry andpattern is designed to shape of the imposed magnetic field to providesensors with a deep depth of sensitivity to the test materialproperties. Near the sensor surface the magnetic field, in thequasistatic limit, decays essentially exponentially, at a ratedetermined by its spatial wavelength and the skin depth in the medium.For a deep depth of sensitivity into the test material, it is generallydesirable to maximize the energy in the dominant spatial mode for thesensor, which has the fundamental spatial wavelength. Drive windingdesigns that gradually taper the current densities to zero near the endsof the winding structure tend to have less energy in the higher orderspatial modes than designs that have abrupt changes or discontinuitiesin the current density at the ends. In one embodiment, drive-windingdesigns that gradually reduce the current density to zero near the endsof the winding structure reduce sensitivity to the higher order spatialmodes, which have shallower penetration depths. Alternatively, inanother embodiment, drive-winding designs that have abrupt changes ordiscontinuities in the current density at the ends are suitable forsimultaneous measurements of both deep and shallow penetration depths.

In another embodiment, drive winding geometry and pattern can bedesigned to minimize sensitivity to unmodeled effects distant from thetest material. In general, unmodeled effects in a sensor response can becompensated by proper sensor calibrations as long as the unmodeledeffects are constant, independent of time and experimental conditions.However, unmodeled effects that are not constant cannot be eliminatedwith sensor calibration which, in turn, leads to a lack of measurementreproducibility. One source of these unmodeled effects is the presenceof magnetizable or conducting objects in the vicinity of the sensor,which may cause the sensor response to change with time as the objectsare moved or when the sensor is moved to new locations or scanned acrossa material. A factor that affects the sensor sensitivity to theseobjects is the decay rate of the magnetic fields around the sensor. Farfrom the sensor the magnetic field decays at a rate determined by thelowest order multi-pole moment excited by the sensor. Conventionalshaped waveform sensors have a dipole moment and a relatively slow fielddecay rate with distance. In one embodiment, the drive winding patternis designed so the dipole moments of each individual loop in the drivewinding cancel. This permits faster field decay rates and lesssensitivity to the extraneous presence of magnetizable and conductingmaterial.

In another embodiment, methods for providing more than one spatialwavelength of the magnetic field excitation are described. Since thedepth of sensitivity of the sensor depends on the dominant imposedspatial wavelength, the use of more than one excitation wavelength willprovide information about the test material property variation withdepth. The use of more than one wavelength within the same footprint hasthe additional advantage that the lift-off (distance between sensor andmaterial) will be the same for both modes of operation. This isparticularly useful when there are more than two unknown parameters andmeasurements under two different applied magnetic field distributionsare necessary to determine all unknowns uniquely.

The fundamental or dominant wavelength of the current excitation can bealtered without changing the geometry of the sensor by changing thecurrent distribution in the segments of the drive winding. In oneembodiment, this is accomplished by supplying an independent currentdrive for each segment and changing the relative magnitudes of thesedrives to create the desired field distribution. For example, thecurrent distribution in the segments may follow a sinusoidal envelopefunction in Cartesian coordinates (or Bessel function in cylindricalgeometry) of the desired spatial wavelength. In another embodiment, adesign that avoids the need for multiple drive circuits uses two or moreindependent winding circuits to create the drive winding segments.Switching the relative current directions between the winding circuitscan then alter the distribution of currents in the winding segments andthe fundamental spatial wavelength for the current excitation. As anexample, consider a drive having two separate winding circuits. In somesegments of the drive, the current through the windings from eachcircuit are in the same direction while in other segments the currentthrough the windings are in the opposite direction. This creates onespatial distribution for the net current in the drive segments. When thecurrent through one of the drive circuits is switched to the oppositepolarity, the spatial distribution of the current in the drive segmentsis also altered.

These method for designing the drive winding structure for sensors thatimpose a shaped magnetic field distribution over a finite sensorfootprint are applicable to non-periodic sensors or sensors with a smallnumber of periods, typically one to four, that are not wide enough to bemodeled accurately assuming unending periodicity. Practical windingdesigns, in both cartesian and cylindrical formats, balance thecompeting constraints imposed by the waveform shaping criteria. Thesewinding structures are suitable for use with a variety of sensingelements for the magnetic field, including standard eddy current coils,Hall effect sensors, magnetoresistive elements, SQUIDS, and hybridcombinations of these elements.

In addition, when the goal is to discriminate between near-surface anddeep material properties, multiple sensing elements can be placed acrossthe footprint of the drive winding structures. In one embodiment, alinear array of sensing elements, at least one of which contains a GMRsensor and feedback coil, are placed adjacent to a linear segment of thedrive winding. This facilitates the creation of property images as thearray is scanned in a direction perpendicular to the array orientation.In another embodiment, a second linear array is placed parallel to thefirst array of sensing elements and offset by half a sensing elementdimension in a direction parallel to the array orientation. This ensuresthat no areas are missed when the array is scanned over a material. Thedistance between the linear arrays and the linear drive segments, whichaffects the depth of sensitivity of the sensor response, can be adjustedto increase the sensitivity of a particular measurement. In anotherembodiment, a two-dimensional array of sensing elements, at least one ofwhich contains a GMR sensor and a feedback coil, can be distributedthroughout the footprint of the drive winding. In an embodiment wherethe drive winding has a cylindrical symmetry, the array of elements cango around the circumference of one of the drive windings. In addition,the shape of the sensing element coils or the feedback coil can beirregular and designed to match the spaces in the drive winding segmentsand the scan pattern. For arrays that use more than one GMR sensor,feedback coils can be wrapped around each individual GMR sensor orgroups of GMR sensors can be encircled by a single feedback coil. Also,since the GMR sensor is sensitive to the orientation of the magneticfield, one or more of the GMR sensors can be oriented differently thanthe other sensors to provide sensitivity to other field orientations.

In another embodiment, a capability for edge detection and correction ispossible with drive windings having linear drive segments by placingsecondary elements both at the center of the primary and near theend-most winding segments. Since the field near the center of theprimary is not significantly different for the infinitely periodicsensor versus the finite width sensor, the sensing elements near thecenter will be insensitive to the presence of the edge until the edge issubstantially beneath the drive. In contrast, sensing elements placednear the edge respond to the fringing fields that are the first to beperturbed by the edge as the sensor is scanned off of the edge.

Another aspect of this invention is the realization that the relativemotion between the magnetoquasistatic sensor and the material under testinfluences both the magnetic field distribution and the sensor response.The relative velocity changes the effective frequency of the excitationand alters the depth of penetration of the imposed magnetic field intothe material under test.

In another embodiment, methods are described for determining thematerial properties from the measured values. In one method, twoproperties or parameter values associated with a measurement aredetermined from measured responses by performing a two-dimensionalinverse interpolation. This interpolation is performed by searchingthrough a database of responses, which can be visualized as grids,locating the grid cell that contains the target or measured point,identifying the parameter values associates with the edges of this gridcell, and performing a bilinear interpolation to determine the finalvalues. In another method, the parameters are estimated usingconventional least-squares minimization technique with the forward stepof the calculation replaced by a table look-up procedure using aprecomputed multi-dimensional database of responses. Since table look-upprocedures are generally very fast compared to simulated responsecalculations, this permits real-time determination of the propertiesfrom the measurement values.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features and advantages of theinvention will be apparent from the following more particulardescription of preferred embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingthe principles of the invention.

FIG. 1 is a typical transfer characteristic of a GMR magnetic sensorbiased at 5 volts.

FIG. 2 shows the structure of the hybrid sensor feedback loop.

FIG. 3 shows a schematic for a feedback and interface circuit.

FIG. 4 shows the structure of a shaped field drive winding in cartesiancoordinates.

FIG. 5 shows the structure of a rotationally symmetric shaped fielddrive winding.

FIGS. 6A, 6B and 6C Equipotential surfaces, Ψ(r, θ, φ)=Ψ₀, of the scalarmagnetic potential for multipole moments with no φ dependence.

FIG. 7 Magnetic field lines for l=3 “octopole” moment potential.

FIG. 8 Number of winding turns for the rectangular sensor with no netdipole moment, according to Table 1, the sinusoidal envelope functionbeing shown with a dashed line, and the number of turns not fallingexactly on the curve, since they need to be integers.

FIG. 9 shows a winding pattern for the circular magnetometer thatenables two different fundamental wavelengths, determined by thepolarity of the connection, with the filled circles corresponding tolong wavelength operation, and the hollow circles corresponding to shortwavelength operation.

FIG. 10 shows conductivity/lift-off measurement grid for circular sensorat 12.6 kHz.

FIG. 11 shows results of conductivity/lift-off measurements with thecircular magnetometer.

FIG. 12 shows a two wavelength magnitude/magnitude permeability/lift-offgrid for a circular sensor with DC excitation.

FIG. 13 shows the results of permeability lift-off measurements with thecircular GMR magnetometer and a 1 mm thick magnetizable foam layer.

FIG. 14 shows a thickness/lift-off measurement grid for the circularmagnetometer at 12.6 kHz. The thickness is of a stainless steel layer onan infinitely thick copper substrate.

FIG. 15 shows the results of a stainless steel layer thicknessmeasurement at five different lift-offs.

FIG. 16 shows a low frequency 100 Hz conductivity/thickness measurementgrid and results for six metal plates.

FIG. 17 shows an expanded view of the upper left corner of the grid inFIG. 16, illustrating the curl in the grid.

FIG. 18 shows a further expanded view of the curl in the upper leftcorner of the grid in FIG. 16.

FIG. 19 shows an area scan of a stainless steel plate with the crack atthe surface.

FIG. 20 shows an area scan of a stainless steel plate with crack 3.2 mmbelow the surface, The position of the crack being indicated with awhite line.

FIG. 21 shows an area scan of a stainless steel plate with crack 7.2 mmbelow the surface. The position of the crack is indicated with a blackline. Because of the change in phase of the induced current, at thisdepth the polarity of the crack signature is reversed

FIG. 22 shows a linear array of sense elements across drive windingshaving Cartesian geometry.

FIG. 23 shows a pair of linear arrays of sense elements across drivewindings having Cartesian geometry.

FIG. 24 shows an areal array of sense elements across drive windingshaving Cartesian geometry.

FIG. 25 contains a circular array of sense elements around drivewindings having a rotationally symmetric geometry.

FIG. 26 shows multiple GMR sensors placed within a feedback coil and atthe center of a drive winding.

FIG. 27 shows multiple GMR sensors placed within a feedback coil andoffset near an edge of a drive winding.

FIG. 28 shows two linear arrays of GMR sensors placed within feedbackcoils and external to the drive winding.

FIG. 29 shows multiple GMR sensors placed within a feedback coil and atthe center of a shaped field distributed drive winding.

FIG. 30 shows a plot of the transinductance variation of a spatiallyperiodic field eddy current sensor as the convection velocity is varied.

FIG. 31 shows a plot of the real part of the magnetic vector potentialfor stationary and moving media.

FIG. 32 shows a schematic diagram of a two-dimensional inverseinterpolation within a grid cell.

FIG. 33 shows a diagram of a grid cell and additional points in acomplex two-dimensional inverse interpolation method.

FIG. 34 illustrates the transformation of a grid cell into aparallelogram.

FIG. 35 illustrates the point and segment definitions for a triangularcell.

FIG. 36 is a flow chart depicting the operations of the database ofresponses and measurement grids.

DETAILED DESCRIPTION OF THE INVENTION

A description of preferred embodiments of the invention follows.

Traditional eddy current sensing devices have a limited capability forlow frequency measurements where sensitivity to deep property variationsare required. These limitations can be overcome by replacing thesecondary winding with alternative secondary elements, such as a Halleffect sensors, magnetoresistive sensors, SQUIDS, and others, whichrespond directly to the magnetic flux density. As described in U.S. Pat.No. 5,793,206, the contents of which are incorporated herein byreference in its entirety, these alternative secondary elements can beincorporated into sensors having meandering drive windings. In oneembodiment, a GMR magnetic field sensor is used. Although magneticsensors based on the magnetoresistive effect are passive components withnon-linear transfer characteristics and limited dynamic range, they donot have a lower bound on the operating frequency and are thus ideal forlow frequency measurements. Indeed, dc or zero frequency measurementsare possible with these giant magnetoresistive sensor elements. Theshortcomings of the magnetic sensor are overcome by biasing into alinear operating regime and placing the sensor into a feedback loopconfiguration with a coil or secondary winding surrounding the magneticsensor and driven to maintain a constant magnetic flux density at thesensor. This feedback loop increases the dynamic range of the sensor andthe amplitude of the magnetic field that can be measured is only limitedby the current driving capability of the associated feedback circuit.While the feedback loop reduces the operational frequency range for theGMR sensor, the GMR sensor with feedback loop allows operation over awider frequency range than a single sensing coil. High frequencymeasurements with the hybrid structure can be performed by eitherrunning the magnetic sensor open-loop, with no feedback, or by measuringthe response of the secondary winding directly, as with a traditionaleddy current sensing device.

The inspection of ferromagnetic materials also motivates the developmentof sensors that can operate at low excitation frequencies. With thesematerials, local magnetic permeability variations are present as theresidual stress varies throughout the material. This complicatesmeasurements of the electrical conductivity of the material since thematerial conductivity and permeability cannot be independently measuredat high frequencies. Lower the excitation frequency permits isolatingthe permeability variations from the conductivity variations.

Some progress has been made in adapting GMR sensors to non-destructivetesting applications. Wincheski and Namkung have integrated a GMR with aself nulling probe driver coil to produce the Very Low Frequency (VLF)Self Nulling Probe. They have operated this device at excitationfrequencies down to 135 Hz and have used it to detect an EDM notch at adepth of up to 10 mm (Wincheski, 2001). Others have attempted this aswell by using conventional eddy current sensor designs (e.g., Rempt,2001). These single element GMR sensors have demonstrated a capabilityto detect cracks under fastener heads and deep into lap jointstructures. The inventions described here address both improved singlesensing element designs and arrays with improved drive winding designs.In an embodiment of this invention, micro-fabricated arrays (or arraysof commercially available GMRs) are incorporated into drive windingdesigns to provide a high resolution, wide area imaging capability.Unique drive coil designs, both square wave and smoothly varying, thatprovide deep penetration while retaining geometric features that permitaccurate modeling of field interactions, combined with high-resolutionarrays of GMR sensors, permit imaging of sub-surface damage andgeometric features.

Despite this progress, there are several complications associated withthe use of GMR sensors in an eddy current probe. They have a highlynonlinear transfer characteristic. The nonlinearity is, in fact, veryextreme, since the sensors are insensitive to the polarity of themagnetic field (see FIG. 1). For this reason, they are typically biasedto an appropriate operating point by an independent magnetic fieldsource, such as a permanent magnet or solenoid. Since the magnetic fieldeffects a resistivity change, the sensors also need electrical biasingand an appropriate bridge configuration. Many of these issues can beaddressed by placing the sensor in a feedback loop, as illustrated inFIG. 2.

FIG. 1 is a plot of the transfer characteristic of a commerciallyavailable magnetic sensor that has been used as a sensing element in aneddy current type sensor. The GMR sensor, manufactured by NonvolatileElectronics, Inc., consists of four magnetic superlattice resistors,deposited on a common ceramic substrate, and connected in a bridgeconfiguration. Two of the resistors are shielded from external magneticfields while ferromagnetic flux concentrators are used to increase theeffect of the magnetic field on the other two resistors. Severalfeatures of the giant magnetoresistive effect are evident in FIG. 1.First, the differential output voltage is independent of the polarity ofthe applied field, because either polarity leads to a departure from thefully antiparallel alignment. Second, at a certain field magnitude,about 50 Oe for this sensor, the output reaches a saturation value,corresponding to complete alignment of the ferromagnetic layers. Andfinally, the response shows some hysteresis.

The highly nonlinear nature of the transfer characteristic, especiallynear zero, makes it necessary to operate the sensor with a DC fieldbias, moving the operating point to the linear region of thecharacteristic. The biasing may be accomplished with a permanent magnet,but this has the undesirable side effect that, in addition to generatingthe constant bias field, it will perturb the field being measured.Alternatively, the sensor may be biased with an electromagnet. While DCbiasing addresses the nonlinearity and hysteresis problems, a severelimitation still remains. Satisfactory operation is maintained only forfields with magnitudes that are small compared to the DC bias field. Asthe field magnitude increases, the response becomes more and morenonlinear, and for amplitudes on the order of the bias field, a polarityreversal occurs.

To address these limitations, the sensor can be placed in a feed backconfiguration with a secondary winding, as shown in FIG. 2 and describedin Sheiretov (2001), the contents of which are incorporated herein byreference in its entirety. In this way the magnetic field at the GMRsensor remains nearly constant during operation, eliminating the effectof the nonlinear transfer characteristic, while maintaining sensitivityat low frequencies. The magnitude of the current in the secondarywinding is taken as the output signal, and since the relationshipbetween this current and the magnetic field for an air-core winding islinear, so is the transfer characteristic of the entire hybrid sensorstructure. The magnetic field magnitude that this hybrid GMR sensor canmeasure is limited only by the magnitude of the field that the secondarywinding can produce, which can be orders of magnitude higher than thesaturation field of the GMR sensor. This dramatically increases thedynamic range of the GMR sensor and makes it's use far more practicalthan in alternative implementations with permanent magnets orelectromagnets that provide a constant bias.

Another benefit of the feedback configuration is temperature stability.Since the measured quantities are currents in the windings, which aredirectly related to the magnetic fields, temperature dependence of theGMR sensor on winding resistance, etc. has no effect on the magnetometerresponse. This is critical since temperature variations have limitedreproducibility and limit the use of many commercially available eddycurrent arrays. Goldfine and Melcher (U.S. Pat. No. 5,453,689, thecontents of which are incorporated herein by reference in its entirety)solved the temperature sensitivity problem for inductive sensingelements by maintaining a gap between drive and sensing windings.Temperature stability is a key to the practical use of GMR sensors aswell.

Another advantage of the feedback connection is for biasing the GMRsensor. Biasing the GMR sensor to the appropriate operating point isaccomplished simply by adding an appropriate DC voltage offset at theinput of the gain stage. This is much better than the alternativebiasing methods described earlier, since correct biasing is maintainedeven if the position of the GMR sensor with respect to the bias sourcechanges, which would not be true for biasing with a constant fieldsource. This eliminates the need for complex alignment methods, sincebiasing at the correct level is automatic with the appropriate choice ofcircuit components. As a result, this feedback configuration providesthe same sensitivity of a GMR sensor by itself while maintaining alinear transfer characteristic and a wider dynamic range.

There can be situations where this kind of feedback configuration may beinappropriate. The bandwidth of the hybrid sensor is limited by thedynamics of feedback loop, and is always narrower than open-loopoperation. Therefore at high frequencies the sensor may need to beoperated in an open-loop mode, although it is still possible to keep theloop closed at low frequencies in order to maintain the proper DC bias.This is not a severe limitation, since this magnetometer is not designedfor high frequency operation, where it may be more appropriate to use astandard spatially periodic field eddy current sensor. An additionallimitation of the feedback loop configuration is that in an arrayarrangement, the feedback windings of adjacent elements will be coupledto each other, potentially leading to inter-channel cross talk. It isgenerally possible to compensate for this in the models for the sensorresponses, but it is better to avoid such situations by appropriateshielding or by open-loop operation.

A full schematic of the electronic circuit used to implement thefeedback loop, biasing, and interface, is shown in FIG. 3. The resistorbridge of the GMR sensor is shown as U₁. The output impedance of thesensor is relatively high, on the order of 40 kΩ, and in order to avoidparasitic signal and noise pick-up, as well as extra phase shift due tocable capacitance, a buffer amplifier is placed in close proximity tothe sensor. The entire buffer stage assembly XX also implements thesensor DC biasing scheme. The buffering is accomplished with a highbandwidth instrumentation amplifier U₂. The voltage regulator U₃ isneeded to ensure constant sensor voltage and field biasing. The DCoperating point is set by the voltage divider implemented with R₁ andR₂. This first stage also has gain of 10. The loop gain andcompensation, described later, are implemented in the U₄ amplifierstage. The secondary winding is driven by the high power video amplifierU₅, connected as a unity gain follower. Since U₅ is a current feedbackoperational amplifier, it is necessary to include resistor R₉, which isalso used to set the bandwidth of the stage. The current I_(F) throughthe secondary feedback winding is measured as the voltage across aprecision zero-inductance power resistor R₁₀ connected in series. Thelast stage, implemented with U₆, is used as a coaxial cable driver. Italso eliminates the DC component of the signal and brings the outputmagnitude to the optimal input levels of the impedance analyzerinstrumentation. For constant field measurements, which do not requirean impedance analyzer, the output signal is taken at test point 3.

Because of the rectifying effect of the GMR sensor, the overall polarityof the feedback loop depends on the direction of the magnetic field.This means that as the circuit is initially powered up, it may enter apositive feedback mode, which is unstable, leads to the output of U₅saturating at the positive supply voltage level, and renders the systeminoperable. On the other hand, if the correct operating point isestablished first, then the feedback will be negative and the loop willthen remain stable, unless the magnetic field reaches such magnitudethat the driver of the secondary winding cannot keep up. In that casethe loop may again become unstable. This complication is a directconsequence of the GMR sensor transfer characteristic and cannot beavoided. It is therefore necessary to implement some external mechanismwhich would detect the faulty condition and reset the loop by forcingthe output to be near the operating point. A much simpler remedy, usedin this implementation, is to ensure that the current through thesecondary winding is always negative, by connecting a diode D₁ in serieswith it. This resolves the stability problem and ensures that thefeedback loop always recovers to the proper operating point. However, italso introduces a limitation. While the magnitude of the measured fieldis unlimited in the positive direction, in the negative direction it maynot exceed the bias field. Although this compromise is acceptable forthe prototype magnetometer, in a more general setting it may benecessary to incorporate an auxiliary “watch dog” circuit of the kinddescribed above.

Another consideration for the feedback circuit is the stability of thecircuit and the bandwidth for the loop. Ignoring for the moment the U₄stage, the ideal open-loop transfer characteristic consists of a singlepole at ω=R₁₀/L, introduced by the relationship between driver outputvoltage and secondary winding current I_(F). In the ideal case wherethis is the only pole, the loop is always stable and its bandwidth canbe increased indefinitely by increasing the loop gain.

In practice all operational amplifiers have a limited bandwidth andparasitic capacitances always introduce extra phase. One approach tofind the optimal loop gain is to measure the open loop transfer functionexperimentally. This is done by shorting out R₆ and C₁, transforming U₄into a voltage follower, applying a signal at test point 2, andmeasuring the relative magnitude and phase of the voltage at test point1. The input signal must include an appropriate DC offset, needed forproper biasing of the GMR sensor. As expected, the phase angle reaches−45° at about 100 kHz, the frequency of the pole. The phase anglereaches −90° at about 350 kHz, which is chosen as the loop bandwidth,allowing for plenty of phase margin, 90°. The gain of the U₄ amplifierstage is chosen to be equal to the inverse of the open-loop magnitude atthis frequency. The presence of C₁ in series with R₆ is useful, thoughnot required, because it introduces a pole at ω=0 and a zero at ω=1/(R₆C₁) near 2 kHz. As a consequence the feedback loop error, given by thevoltage at test point 1, is zero at DC and small at low frequencies.

When modeling the response of the sensor in this feedback loop, thefield generated by the current in the feedback winding must beconsidered. One cannot assume that the output signal I_(f) is directlyproportional to the imposed field at the origin since the secondarywinding is a solenoid tightly would around the GMR sensor. Experimentsshowed that this assumption is not justified since the field created bythe secondary winding is influenced by the proximity and properties ofthe material under test. As a result, the net field imposed on the GMRsensor at the origin, created by the drive winding and the currentthrough the secondary winding, must be considered when calculating thetransfer function (ratio of the secondary current to the drive current)for the feedback sensor structure. Normalizing the response of thesensor to a measurement in air, as with a standard air-calibrationmeasurement procedure, then allows the absolute properties of thematerial under test to be determined.

In addition to the frequency, the geometry of the drive winding alsoaffects the depth of penetration of the magnetic field into the testmaterial. Passing a current through the drive winding creates a magneticfield that has spatial modes determined predominantly by the geometry ofthe drive winding. In general, the shorter of the skin depth into thematerial or the wavelength of the dominant spatial mode limits the depthof penetration. Spatially periodic field eddy-current sensors, designedto create a single dominant spatial mode, have been used to measure foilthickness, characterize coatings, and measure porosity, as well as tomeasure property profiles as a function of depth into a part, asdisclosed in U.S. Pat. No. 5,015,951, the contents of which areincorporated herein by reference in its entirety, and U.S. Pat. No.5,453,689. Novel nonperiodic shaped-field drive winding structures, alsodesigned to create a single dominant spatial mode, have also beendisclosed, for example, in U.S. patent application Ser. No. 09/488,241,the contents of which are incorporated herein by reference in itsentirety. For controlling the magnetic field shape, discrete distributeddrive windings are used. Rather than driving each current segmentindependently, one or more continuous winding conductors are used, withthe correct current profile obtained by changing the number of windingturns within each segment.

Shaped field winding patterns for the drive winding can be created in avariety of geometries. As an example, FIG. 4 shows windings comprised ofrectangular loops for a Cartesian coordinate pattern. Terminalconnections to these loops are made through the wires 23 andinterconnections between the loops are made through the wires 24. Thenumber of turns in each segment 22, denoted by the capital letters Athrough K, is varied so that the shape of the cross-sectional currentdistribution in the y direction can be adjusted as necessary. Continuityof the current is maintained with the side connections 20. These sideconnections are typically placed far enough apart so that no variationsin the x direction need to be considered when modeling the sensorresponse. An example winding current distribution designed to excite asingular Fourier mode for the magnetic field is illustrated in FIG. 8and listed in Table 1. For each segment, the sign for each valueindicates the current direction while the integer indicates the numberof conducting segments or relative current magnitude. Another example isthe rotationally symmetric cylindrical geometry FIG. 5, where the numberof turns and current direction in each discrete circular loop 30 isvaried to shape the field. Interconnections between each segment aremade with tightly wound conductor pairs 32 to minimize fringing fieldeffects. A GMR sensor 34, with feedback controlled coil, is placed atthe center of the concentric circular drive windings. Sensor 34 may alsobe a coil, a SQUID sensor, or a Hall effect sensor. Connections to thissensing element are made with a tightly wound conductor pair 36. Boththe number of turns and the polarity of the windings (current direction)can be varied in the drive winding segments. In this case, there are twosets of drive windings (31 and 33), with each loop in the radialdirection aligned with one another and placed nearby so that thedistance in the z direction between the windings and also the senseelement 34 are small compared to the largest loop diameter, which allowsmore than one fundamental spatial mode. As described later, the polarityof the connection 32 determines which of the two current drive patterns(with different fundamental spatial wavelengths) is excited. Thisprovides two distinct field depth of penetration conditions and permitsimproved multiple property measurements for layered media. The capitalletters for each winding indicate the number of turns and the arrows oneach indicate that the winding direction can also be changed betweenloops. The test material 35 can be a substrate 39 having a magnetizablefoam layer 37 of known thickness. An example winding distributiondesigned to excite a singular Fourier-Bessel mode for the magnetic fieldis illustrated in FIG. 9. This example uses the winding turn and currentdistribution listed in Table 3, where the patterns A and B correspond tothe turn distributions of the windings 31 and 33.

Once the sensor response is obtained, an efficient method for convertingthe response of the GMR sensor into material or geometric properties isto use grid measurement methods. These methods map the magnitude andphase of the sensor response into the properties to be determined. Thesensors are modeled, and the models are used to generate databasescorrelating sensor response to material properties. Only by constructingthese measurement grids off-line is it possible to convert to materialproperties in real-time. The measurement grids are two-dimensionaldatabases that can be visualized as “grids” that relate two measuredparameters to two unknowns, such as the conductivity and lift-off (wherelift-off is defined as the proximity of the test material to the planeof the sensor windings). For coating characterization or forinhomogeneous layered constructs, three-dimensional grids (or higherorder grids), called lattices (or hyper-cubes), are used. Similarly, amodel for the GMR sensor with feedback loop and circular drive windingswas developed and used to generate measurement grids, which were thenused to interpret sensor response. Alternatively, the surface layerparameters can be determined from numerical algorithms that minimize theleast-squares error between the measurements and the predicted responsesfrom the sensor.

It is also possible to combine the grid measurement methods with aleast-squares minimization method for a hybrid inverse estimationmethod. In this case, the standard error function minimization techniqueof standard minimization methods would be used. The forward steps of theestimation that require calculations of the response from the modelwould be replaced by forward grid lookup routines evaluated onprecomputed databases of responses. Standard grids could be used fortwo-unknowns, lattices for three unknowns, and higher dimensionaldatabases for greater unknowns. This has the advantage of being muchfaster than the standard minimization algorithm, provides a multipleunknown estimation capability, may incorporate data from manymeasurements taken under a variety of conditions, and works in caseswhen post-processing of simulated results is necessary, such as toaccount for parasitic or otherwise unmodeled effects. The disadvantagesare that the databases need to be generated before hand and the methodis not guaranteed to find a solution. This latter disadvantage is also alimitation of standard minimization techniques.

An advantage of the measurement grid method is that it allows forreal-time measurements of the absolute electrical properties of thematerial. The database of the sensor responses can be generated prior tothe data acquisition on the part itself, so that only table lookupoperation, which is relatively fast, needs to be performed. Furthermore,grids can be generated for the individual elements in an array so thateach individual element can be lift-off compensated (or compensated forvariation of another unknown, such as permeability or coating thickness)to provide absolute property measurements, such as the electricalconductivity. This again reduces the need for extensive calibrationstandards. In contrast, conventional eddy-current methods that useempirical correlation tables that relate the amplitude and phase of alift-off compensated signal to parameters or properties of interest,such as crack size or hardness, require extensive calibrations andinstrument preparation.

One of the other considerations for the drive winding structures is thedecay of unintended or stray fields away from the sensor. These strayfields, and their decay pattern, can be understood qualitatively byconsidering the far-field decay patterns from multipole sources. Forexample, the drive winding may have a net dipole moment such that, atdistances much larger than the sensor wavelength, the magnetic fieldapproaches the form of a field generated by a magnetic dipole at theorigin. The net dipole moment of concentric-drive windings will not bezero if each winding forms a current loop of non zero area, and theeffective magnetic moments of all these loops point in the samedirection.

An illustration of this use of the database of responses and themeasurement grids is shown in FIG. 36. First, properties that are ofinterest are to identified 91. Examples of properties include, but arenot limited to, conductivity, permeability, and lift-off. Next, adatabase of precomputed sensor responses may be generated 93. Thegenerated database may relate obtained measurement values with theproperty values to be estimated. The database may typically be generatedwith use of a model for the sensor response and with the ranges for theproperties of interest that span the application for which the method isbeing applied. This database may be generated prior the measurementvalues and is therefore considered precomputed. After obtaining themeasurement value 95, the property values may be estimated by minimizingthe difference between the measurement values and the database ofprecomputed sensor responses 97.

Near the sensor surface the magnetic field decays essentiallyexponentially, at a rate determined by its spatial wavelength and theskin depth in the medium. Far from the sensor the fields fall off at arate determined by the lowest order multipole moment excited by thesensor. Ideally, this will be a high order moment, since it is desirablefor the sensor to lose sensitivity quickly with distance. Otherwise thesensor response will be affected by the presence of magnetizable and/orconducting objects in its vicinity, even if they are much farther thanthe material under test. For example, internal structural supports mayinterfere with mapping of corrosion in an aircraft lapjoint. Whereasthis effect may be small, even compared to other unmodeled effects, itcan change with time, as objects near the sensor are moved, or when thesensor is moved to inspect a new location. Thus, it cannot be eliminatedvia calibration and will result in a lack of reproducibility.

For magnetic fields, the lowest order multipole solution possible toLaplace's equation in spherical coordinates is the dipole, whose scalarmagnetic potential decays as r⁻². FIGS. 6A, 6B and 6C show theequipotential surfaces of the scalar magnetic potential for the threelowest order rotationally symmetric multipole modes. The “quadrupole”(l=2) mode cannot be excited by these magnetic sensors, since it is evenwith respect to z, whereas flipping the sensor upside down reverses thecurrent direction in all windings, changing the sense of all magneticfields. If the dipole moment is eliminated, the next dominant mode isthe “octupole,” whose magnetic field lines are shown in FIG. 7. Thepotential of this mode has an r⁻⁴ dependence, which means that themagnetic field intensity falls off as r⁻⁵. In practice the decay rate ofthe octupole is fast enough so that canceling out the dipole moment issufficient to eliminate unwanted long range sensitivity.

For the net dipole moment of the drive winding to cancel, thecontributions of each individual winding segment (including selectednumbers of turns to shape the field) must sum to zero. To satisfy thisconstraint, the sensor must include a fraction of a wavelength past theend of the single period. For this additional fraction of a wavelength,the current is in the opposite direction. Since the number of windingturns is always an integer, it is impossible to satisfy the zero-sumconstraint exactly while at the same time strictly following thesinusoidal envelope function. Another consideration in choosing thenumbers tapering off the current towards the ends of the sensor in orderto avoid rapid changes in the primary current distribution, which wouldresult in more energy in higher order Fourier modes, and consequently, areduction of the sensitivity at greater depths. A compromise betweenthese considerations for a shaped field sensor in Cartesian coordinates(drive segments placed at different locations in the y-direction andassuming the segments are long enough in the x-direction so that novariation in the field distribution in the x-direction needs to beconsidered) is shown in Table 1, which lists a winding pattern thatresults in no net dipole moment. The winding turns are also plotted inFIG. 8. The current in the last two segments deviates from the sinusoidbut prevents the generation of the higher order spatial modes.

TABLE 1 Drive winding pattern for a rectangular magnetometer having nonet dipole moment. j = y/h 0 1 2 3 4 5 6 7 8 9 10 11 12 Winding 0 5 9 1213 12 9 5 0 −5 −9 −7 −4 turns ω_(j)

It is also possible to change the fundamental wavelength of the currentexcitation without changing the geometry of the sensor, by changing thecurrent distribution in the primary winding segments. Since the depth ofsensitivity of the sensor depends on the imposed spatial wavelength, theuse of more than one excitation wavelength will provide some informationon how material properties change with depth. Furthermore, the use ofmore than one wavelength within the same footprint has the advantagethat the lift-off (distance between sensor and material) will be thesame for both modes of operation. This is very useful when there aremore than two unknown parameters and measurements under two differentapplied magnetic field distributions are necessary to determine allunknowns uniquely. This may be accomplished by supplying an independentcurrent drive for each segment and changing the relative magnitudes ofthese drives to follow an appropriate envelope function with the neededwavelength. A more practical approach avoids the need for multipledrives by using two or more independent winding circuits and switchingthe relative current direction between them.

As an example, consider the winding distribution of Table 2 for arectangular or Cartesian sensor. Two windings, A and B, and the numberof turns each has in every current segment, are shown in this table. Asbefore, negative turns indicate that they are wound in the oppositedirection. When the two windings are driven in the same polarity, theresulting current distribution excites a mode with a fundamentalwavelength equal to the length of the sensor. If, on the other hand, thetwo windings are driven with opposite polarities, the resultingexcitation has a fundamental wavelength equal to one-half of the sensorlength. This approach can be followed in both Cartesian coordinates andcylindrical coordinates. Table 3 shows the winding pattern for amagnetometer having circular symmetry with the current distributiondesigned to match a first order Bessel function. This distribution isillustrated in FIG. 9, where the filled circles correspond to the A+Bconnection and the hollow circles correspond to the A−B connection. Thecurves show the envelope functions. The number of turns do not fallexactly on the curves because they need to be integers. The number ofturns in the last two windings are tapered off in order to avoid rapidchanges in the primary current distribution. Envelope distributionsfollowing a Bessel function were used since these functions arise whenmodeling the circularly symmetric versions of the magnetometer. Notethat rotationally symmetric magnetoquasistatic and electroquasistaticsensor geometries are suitable for measurements on circularly symmetriccomponents or when a dependence on sensor directionality is undesired.

TABLE 2 Two-wavelength winding pattern for a rectangular magnetometer.Nominal Turns per Segment A 0 14 22 21 13 3 −4 −4 0 4 4 −3 −13 −21 −22−14 0 B 0 −4 −4 3 13 21 22 14 0 −14 −22 −21 −13 −3 4 4 0 A + B 0 10 1824 26 24 18 10 0 −10 −22 −24 −26 −24 −18 −10 0 A − B 0 18 26 18 0 −18−26 −18 0 18 26 18 0 −18 −26 −18 0

TABLE 3 Two-wavelength winding pattern for a circularly symmetricmagnetometer. Nominal Turns per Segment A 13 21 21 15 7 1 −1 0 1 −4 −3 B−3 −3 1 9 15 15 9 0 −9 −4 −3 A + B 10 18 22 24 22 16 8 0 −8 −8 −6 A − B16 24 20 6 −8 −14 −10 0 10 0 0

The shaping of the imposed magnetic field allows the sensors to bedesigned with a deep depth of sensitivity to the test materialproperties and a minimal sensitivity to unmodeled effects distant fromthe test material.

Near the sensor surface the magnetic field, in the quasistatic limit,decays essentially exponentially, at a rate determined by its spatialwavelength and the skin depth in the medium. For a deep depth ofsensitivity into the test material, it is generally desirable tomaximize the energy in the dominant spatial mode for the sensor, whichhas the fundamental spatial wavelength. Drive winding designs thatgradually taper the current densities to zero near the ends of thewinding structure tend to have less energy in the higher order spatialmodes than designs that have abrupt changes or discontinuities in thecurrent density at the ends. To reduce sensitivity to the higher orderspatial modes, which have shallower penetration depths, drive-windingdesigns that gradually reduce the current density to zero near the endsof the winding structure should be used. Alternatively, if simultaneousmeasurements of both deep and shallow penetration depths are desired,then drive-winding designs that have abrupt changes or discontinuitiesin the current density at the ends could be used. In addition, when thegoal is to discriminate between near-surface and deep materialproperties, multiple sensing elements can be placed across the footprintof the array. The trade-off is in the data acquisition across parallelchannels, lead connections, and switching of excitations as described inthe next paragraph. In addition, for the Cartesian geometry, acapability for edge detection and correction by placing secondaryelements both at the center of the primary and near the end-most windingsegments is possible. Since the field calculation models show that thefield near the center of the primary is not significantly different forthe infinitely periodic versus the finite width sensor, the sensingelements near the center will be insensitive to the presence of the edgeuntil the edge is substantially beneath the drive. In contrast,secondaries placed near the edge respond to the fringing fields that arethe first to be perturbed by the edge as the sensor is scanned off ofthe edge.

Practical winding designs, in both cartesian and cylindrical formats,balance the competing constraints imposed by the waveform shapingcriteria. These winding structures are suitable for use with a varietyof sensing elements for the magnetic field, including standard eddycurrent coils, Hall effect sensors, magnetoresistive elements, SQUIDs,and hybrid combinations of these elements.

Several sets of measurements have been performed with a circularlysymmetric shaped field magnetometer. These measurements used the GMReddy current sensor with drive illustrated in FIG. 9. A simple one-pointair calibration method is used for all of these measurements. This meansthat the sensor response when over the test material was normalized bythe sensor response in air, away from any conducting or magneticmaterials. The measurement results are then processed with measurementgrids to provide absolute property measurements, such as electricalconductivity, magnetic permeability, material thickness, and sensorproximity (lift-off). The absolute property measurement capabilityeliminates the need for extensive, and in some cases any, calibrationsets. Even if reference calibrations are performed, possibly to improvethe accuracy of the property estimation, only a single calibrationmaterial may be required. Air and reference part calibration methodshave previously been described for square wave meandering windingconstructs in U.S. Pat. No. 6,188,218, the contents of which areincorporated herein by reference in its entirety. The discrete segmentCartesian and circular geometry sensors described herein can also becalibrated in this fashion because the sensor response can be accuratelymodeled. In principle, air calibrations in this context can be performedwith any sensor whose response can be accurately modeled.

FIG. 10 shows the measurement grid for conductivity/lift-offmeasurements with three different materials, in the form of metalplates, over a range of lift-off values. Since both the conductivity andthe lift-off parameters vary over a relatively large range, theparameter values for this grid are chosen on a logarithmic scale. Thegrid cell area is a measure of the sensitivity of the measurement inthat region of the grid. The measurements are carried out at 12.6 kHz.Placing plastic shims between the sensor and the metal plates varied thelift-off.

The results are shown in FIG. 11. The three data sets follow lines ofconstant conductivity very closely. As listed in Table 4, the measuredlift-off values were in excellent agreement with the nominal values.Only the first 12 sets are listed, due to the lack of sensitivity athigher lift-off values, as illustrated by the narrowing of the gridcells in FIG. 11.

The lowest value of the lift-off, 3.3 mm, corresponds to measurementswith no shim, and is equal to the effective depth of the windings belowthe surface of the sensor. This amount has been added to the data in thelast column, after having been estimated by taking the average of thedifference between the magnetometer estimated values and the measuredshim thicknesses. This number is quite reasonable, given that theaverage depth of the grooves is on the order of 3 mm, and that thewinding thickness, about 2 mm, is not considered by the model. Theconductivity data in Table 4 are also in good agreement with valuesreported in the literature. There appears to be an optimal range of thelift-off, 5-7 mm, where the estimated conductivity is most accurate.This is reasonable since sensitivity is lost at higher lift-offs, whilea close proximity to the sensor windings is also not desirable since theeffects of the non-zero winding thickness then become more significant.These conductivity results are also remarkable good considering thatthis measurement was carried out with no calibration standards and witha single air calibration point, the model for the sensor response isrelatively simple, and no empirical data have been used to determine thesensor response. If it is necessary to perform a very exact conductivitymeasurement, then a two-point reference part calibration is recommended,with the properties of the two reference parts (or the same part at twolift-off values) bracketing the properties of the unknown part.

TABLE 4 Measurement results corresponding to FIG. 11. Nomi- nal DataConductivity [MS/m] Lift-off [mm] Lift-off Set Cu 110 Al 6061 Al 2024 Cu110 Al 6061 Al 2024 [mm] 1 59.2 29.5 18.0 3.2 3.3 3.3 3.3 2 59.2 28.917.8 4.0 4.1 4.1 4.1 3 58.7 28.7 17.8 4.7 4.8 4.5 4.8 4 58.3 28.6 17.65.5 5.6 5.6 5.6 5 57.8 28.3 17.6 6.4 6.5 6.5 6.5 6 57.1 28.1 17.5 7.37.1 7.3 7.3 7 55.7 27.4 17.3 7.9 8.0 8.0 8.0 8 56.1 27.5 17.4 8.7 8.98.8 8.8 9 54.3 26.8 17.1 9.4 9.5 9.4 9.4 10 55.2 27.0 17.2 10.2 10.310.3 10.2 11 53.5 26.4 17.0 10.8 10.9 10.9 10.9 12 53.0 26.3 16.7 11.711.7 11.7 11.7

These results confirm the validity of the model for this cylindricalcoordinate sensor. Additional measurements were made to illustrate theunique advantages of the GMR magnetometer by using frequencies andmaterial thicknesses outside the range of standard eddy current sensors.

One example measurement illustrating the unique capabilities of thecylindrical GMR based magnetomer is the use of two-different spatialwavelengths and a DC excitation to determine the permeability of a layerof known thickness and it's lift-off from the sensor surface. This typeof measurement could apply, for example, to the measurement of theproximity and permeability of a low observability (LO) coating, such asa magnetizable foam layer. This type of measurement is unique in threeimportant ways: (1) quasistatic sensors are generally not operated inthe fully static regime at DC, (2) two-wavelength magnetometry has neverbeen demonstrated, and (3) the wavelength of the sensor can be changeddynamically, with the material under test still in place. These are alluseful capabilities, especially in measuring magnetizable materialswithout the limitation caused by the skin depth effect.

With DC operation the diffusion term in the differential equationsdisappears and the problem becomes completely parallel to dielectrometrymeasurements on insulating materials. All amplitudes in the models forthe sensor response become real numbers (as opposed to the complexnumbers that are used when the frequency is non-zero) and onemeasurement provides only one degree of freedom for determining theproperties of the material under test. A magnitude/magnitude measurementgrid used for parameter estimation is shown in FIG. 12. The spiralnature of the grid is due to the influence of the material under test onthe feedback loop transfer function. This behavior is explained asfollows: In general, as a magnetizable material is brought closer to thesensor windings, the magnitude of the signal increases, because thematerial tends to concentrate the magnetic flux. However, as thelift-off is reduced even further, this field concentration effect beginsto affect the secondary feedback winding too, and since its radius ismuch smaller than the radii of the segments of the primary winding, thiseffect begins to dominate the overall magnetometer transfer function,reducing the magnitude.

This type of measurement is completely analogous to the dielectrometermeasurements where the permittivity of a material is to be determined,as opposed to the permeability of a material. Both types of measurementsuse the signal magnitudes at two different spatial wavelengths toestimate the permittivity or permeability of a sample with knownthickness, and its distance from the sensor electrodes or windings. Themagnetizable layer consists of 1 mm thick polymer, containingferromagnetic particles suspended in the polymer foam.

Representative measurement results, with the lift-off varied by placingplastic shims between the material under test and the sensor, are shownin FIG. 13, and listed in Table 5. Performing the measurements at DCrequired some special procedures. On the one hand, the only necessaryequipment is a power supply and a voltmeter so that the measurement isrelatively simple. On the other hand, the accuracy of the measurement islower than that possible with AC excitation. When operating the sensorat a specific frequency, the magnitude of this frequency component inthe output signal can be measured very accurately by mixing it with asignal at the same frequency from the same source and passing theresulting signal through a low pass filter. The resulting DC componentis then related to the magnitude of the output signal. The signal can beintegrated over many cycles, which results in a very accuratemeasurement of the transfer function at this frequency. With DCoperation this frequency mixing technique is not available. Furthermore,it is not practical to average the signal over a long time, becauseoutside factors affecting the measurement may shift during that time.For example, the earth's magnetic field is responsible for part of thesignal and its value changes during the day due to the changing state ofthe ionosphere and to the contribution of the magnetic field of the sun.Another complicating factor is that the primary current may changethrough instabilities in the output voltage of the power supply, changesin the contact resistance of the leads, and temperature induced changesin the resistance of the primary winding itself as the equilibriumtemperature varies. Also, changes in the physical position of the sensorcan alter its output due to the presence of magnetically active objectsin the vicinity and to changes of its position relative to the directionof the earth's magnetic field. The approach taken in this measurement toeliminate the signal noise introduced by all these factors is to measurethe sensor response in air not just once before the measurement butbefore every data point, with as little time in between as possible. Inthis way every measurement data point has its own air calibrationreference.

TABLE 5 Permeability/Lift-off measurement results corresponding to FIG.13. Nominal Data Relative Lift-off Lift-off Set Permeability [mm] [mm] 14.23 3.3 3.3 2 4.30 4.1 4.1 3 4.30 4.8 4.8 4 4.29 5.7 5.6 5 4.28 6.6 6.56 4.19 7.2 7.3 7 4.21 8.0 8.0 8 4.36 9.1 8.8 9 4.37 9.9 9.4 10 4.39 10.810.2 11 4.21 11.1 10.9 12 4.28 12.2 11.7 13 4.43 13.5 12.6 14 4.44 14.313.4 15 4.41 14.9 14.1 16 4.50 16.1 14.9

Despite the limitations associated with performing DC measurements, theresults in Table 5 are still good. The lift-off tracks the nominallift-off values, but not as well as the data of Table 4. The relativepermeability of the polymer layer is measured to be about 4.3 and thedata follow a constant permeability curve quite well. As in theconductivity/lift-off measurements of Table 4, the measurement resultsare most accurate for lift-off values in the 5-7 mm range. It is alsonoteworthy that the, in FIG. 13, that at the highest lift-off values,there is a range where the magnitude of the short wavelength signal islower than that of air. The grid lines confirm that this effect isphysical. It is due to the fact that far away from the surface themagnetizable layer is too distant to intercept any magnetic field linesgoing through the center of the sensor but is still sufficiently closeto attract some of the field lines away from the surface. An exact dualof this effect is observed in the dielectrometer grids in the presenceof a ground plane behind the material sample.

Many applications often require the measurement of the thickness of ametallic coating on a metal substrate. One typical multi-layer structurehas at least three layers, two metal layers and an air layer, where theconductivity and magnetic permeability of the two metal layers areknown. Low frequency operation, made possible by the GMR eddy currentsensor, will allow measurement of much higher coating thicknesses thanexisting spatially periodic field eddy current sensors.

The measurement grid used in this application is shown in FIG. 14, whichassumes a stainless steel layer is on an infinitely thick coppersubstrate. Note that a material can be considered infinitely thick whenthe thickness is several times larger than the depth of penetration ofthe magnetic field into the material. For generating the grid, theconductivity of the stainless steel was taken as 1.39 MS/m and theconductivity of the copper was taken as 58 MS/m. The copper plate is 3.2mm thick, but is modeled as infinite, since it is several times greaterthan the value of the skin depth of copper at this frequency (0.59 mm).In comparison, due to its much lower conductivity, the skin depth in thestainless steel layer (3.8 mm) is several times greater than that of thecopper layer, and comparable to the plate thickness. By considering thesize of the grid cells, it can be observed that the thicknessmeasurement (but not the lift-off) loses sensitivity for thicknessvalues below about 0.5 mm and above 5 mm. This is explained by comparingthese values to the skin depth of 3.8 mm. At the lower thickness valuesthe stainless steel layer has little influence on the magnetic fields,since its conductivity is much lower than that of the copper layer,which dominates the sensor response. At the high end of the thicknessrange the exponential decay of the magnetic field intensity makes thesensor insensitive to the position of the interface between thestainless steel and the copper. For these two limits the grid linesasymptotically approach two constant lift-off lines of theinfinite-half-space conductivity/lift-off rid in FIG. 10, correspondingto the conductivities of copper and stainless steel. Optimalsensitivity, corresponding to the most open and orthogonal grid cells,is achieved for thicknesses on the order of 3 mm, close to δπ/4.

The thickness of the stainless steel layer in this set of measurementswas changed by stacking up to four plates of various thicknesses indifferent combinations. Twelve sets of data were taken at each of fivedifferent lift-off values. The results are shown in FIG. 15 and listedin Table 6. Excellent agreement between the nominal and estimated valuesof thickness and lift-off was obtained, which confirmed the validity ofthe analytical model including representation of field interaction witha multiple layered material under test. Higher frequency measurementswould improve the sensitivity to the thickness of thinner coatings byreducing the skin depth.

TABLE 6 Coating thickness estimation results corresponding to the dataof FIG. 15. h = h = h = h = h = Data 3.3 mm 4.1 mm 4.8 mm 5.6 mm 6.5 mmNominal Set Thk. Lift. Thk. Lift. Thk. Lift. Thk. Lift. Thk. Lift.Thick. 1 −0.33 3.57 −0.33 4.40 −0.33 5.05 0.02 5.60 0.24 6.33 0.00 20.61 3.26 0.63 4.06 0.63 4.72 0.71 5.54 0.69 6.49 0.60 3 0.92 3.26 0.934.09 0.94 4.81 0.94 5.64 0.97 6.54 0.96 4 1.49 3.23 1.50 4.08 1.56 4.731.53 5.57 1.55 6.51 1.50 5 1.93 3.23 1.94 4.07 1.95 4.76 1.95 5.60 1.976.53 1.89 6 2.14 3.21 2.16 4.06 2.16 4.73 2.15 5.60 2.16 6.51 2.10 72.57 3.25 2.58 4.06 2.56 4.78 2.56 5.63 2.57 6.54 2.49 8 2.9 3.24 2.904.06 2.88 4.77 2.89 5.60 2.90 6.53 2.85 9 3.45 3.24 3.47 4.05 3.50 4.723.50 5.55 3.52 6.49 3.39 10 4.13 3.22 4.12 4.04 4.13 4.75 4.13 5.55 4.106.50 3.99 11 4.33 3.24 4.43 4.06 4.46 4.75 4.46 5.55 4.44 6.49 4.35 125.03 3.23 5.01 4.05 5.30 4.76 5.04 5.55 5.04 6.49 4.95

Another example application is thickness measurements of relativelythick aluminum plates, where the thickness can be 0.25 inches orgreater. This requires the use of low excitation frequencies, such as100 Hz, where the skin depth is 9.5 mm in aluminum. One importantapplication of this type of measurement is corrosion mapping, where thesurface that corrodes is often not accessible for direct measurement.Since the other material property being measured is conductivity, a scanof this type would simultaneously detect cracks and other flaws in themetal. Also, multiple frequency methods could be added to vary the fielddepth of penetration.

The conductivity/thickness grid used in this set of measurements isshown in FIG. 16. In this case, this grid does not have lift-off as oneof the two estimated parameters. The lift-off is assumed to be known andequal to 3.3 mm, the intrinsic magnetometer value obtained from earliermeasurements. There are two reasons why lift-off is almost always one ofthe unknown properties in magnetometer measurements: (1) the lift-off isusually not known, because at the lower thickness scales typical of eddycurrent sensors, dust particles and surface roughness make anon-negligible contribution; and (2) grids including lift-off have moreclosely “orthogonal” cells, as other material properties are lessindependent of each other when they enter the model. The relativelylarge minimum lift-off value for this sensor reduces the importance ofthe first consideration. The second consideration still remains as anissue, as can be observed in the upper right corner of the grid wherethe grid cells collapse and approach zero area, as their edges becomealmost parallel to each other. Grid look-ups in such an area of the gridare naturally unreliable. Nonetheless, there is a big area of the gridwhere the two unknown properties are sufficiently independent of eachother.

An interesting property displayed by the grid of FIG. 16 is that nearthe upper left corner, lines of constant conductivity form spirals asthey approach the limiting point corresponding to infinite thickness. Asa result, the grid folds in on itself, sometimes several times for alarge range of thicknesses. This means that for certain values of thecomplex sensor magnitude there can be two or more solutions, allphysically valid. Expanded views of this region of the grid are shown inFIG. 17 and FIG. 18. The grids can manifest this kind of behavior whenone of the unknown parameters is the thickness of the metal layer. Thisis caused by the fact that in the presence of magnetic diffusion, theexponent of the z-dependent term is a complex number and, due to theimaginary part of the exponent, the phase of the induced eddy currentschanges with depth. As a result, the missing tail of the exponent, dueto the finite width of the layer, can alternatively enhance or reducethe fields at the secondary sensor, leading to the spiral grid effect.This effect has been observed for other deep penetration eddy currentsensors such as the magneto-optical imaging (MOI). This behavior isanalogous to the evanescent decay of standing waves of electromagneticfield in high frequency (e.g., microwave or optical) systems such astransmission lines or the reflection and transmission of plane waves atinterfaces.

The thick plate measurement results, listed in Table 7, show goodagreement with the conductivity values in the literature and the nominalthicknesses measured with a caliper. One notable exception is the 1.3 mmthick aluminum plate. This plate thickness is simply out of the range ofsensitivity for this measurement, as can be seen visually in FIG. 16,where this point falls in a very narrow region of the grid. Similarly,the 2024 Al 3.2 mm point is also in a very narrow region, but actuallyresulted in a good thickness and conductivity estimate for this example.

TABLE 7 Low frequency (100 Hz) conductivity/thickness measurementresults corresponding to the data of FIG. 16. Nominal MeasuredConductivity Thickness Conductivity Thickness Data Set Material [MS/m][mm] [MS/m] [mm] 1 Cu 110 58.0 3.2 56.2 3.39 2 Al 2024 17.5 3.2 17.53.34 3 Al 2024 17.5 1.3 12.5 2.19 4 Al 6061 27.3 6.7 29.1 6.53 5 Al 202417.5 9.9 17.1 10.38 6 Al 6061 27.3 9.8 28.1 9.52

Another set of measurements illustrates the GMR magnetometer capabilityto detect material flaws in a thick layer of metal. These measurementswere carried out by performing scans over a set of stainless steelplates. One plate had a 25 mm long, 0.4 mm wide, and 2.4 mm depth slotto simulate a crack. The grid used for this measurement is theconductivity/lift-off grid in FIG. 10. The crack is not modeledexplicitly, but its presence is usually manifested by a local reductionin the value of the measured conductivity. In some cases, depending onits depth and position below the surface, it may appear as a localchange in the lift-off.

Three sets of scans were made with stainless steel plates arranged tosimulate a crack at the upper surface, nearest the sensor, a crack 3.2mm below the upper surface, and a crack 7.2 mm below the surface. Theimage generated by the first scan, with the slot at the surface, isshown in FIG. 19. This image shows the conductivity, normalized by itsvalue away from the crack. The crack signal is very strong, with theconductivity decreasing more than 3% near the crack position. The doublehump signature of the crack is characteristic of the effect cracks haveon the signal of imposed-periodicity eddy current sensors. The inducedcurrent density mirrors the current density of the drive, and as aconsequence, the disruption caused by the crack is greatest when it isdirectly below, and perpendicular, to the primary winding nearest to thesensing element.

The conductivity image generated with the crack positioned 3.2 mm belowthe surface is shown in FIG. 20. The change of the effectiveconductivity is approximately 2.5%. The image generated with the crack7.2 mm below the surface is shown below in FIG. 21. In this case, theconductivity change was less than 0.5%, which is only slightly above thenoise level. This is to be expected, since the skin depth of stainlesssteel at this excitation frequency is 3.8 mm. The quality of the imageat this depth can be improved by measuring at a lower frequency.

An interesting feature to be observed in FIG. 21 is that near the crack,the measured conductivity is actually higher. This is because the phaseof the induced eddy currents changes with depth and is also what causesthe grid in FIG. 16 to curl inwards. With the crack positioned 7.2 mmbelow the surface it interrupts eddy currents that are flowing in adirection opposite to the surface eddy currents, thereby increasing themagnetic field at the sensor. A consequence of this effect is that thereis a characteristic depth, near π/2 skin depths, where a crack wouldcause no change in the conductivity. For this reason, it is commonpractice to test at more than one frequency. This is not as troublesomeas it may seem, for while the real part of the exponent may be zero, atthis characteristic depth the imaginary component is not, and thereforethe crack signature would show up in a plot of the lift-off. This isalso evident in the grid in FIG. 16, where a constant conductivity linemay change its direction along a spiral, but never crosses itself.

A variety of other adaptations for the device design are also available.When the goal is to discriminate between near-surface and deep materialproperties, multiple sensing elements can be placed across the footprintof the array. These arrays can incorporate giant magnetoresistive sensorelements. FIG. 22 shows a linear array 40 of sensing elements 42 placedalong the centerline 44 of a Cartesian array of drive winding segments46. The linear array is perpendicular to the scan direction tofacilitate the construction of images of the measured materialproperties. FIG. 23 shows a pair of linear arrays 50 of sensing elements52 within the footprint of a Cartesian array of drive winding segments54. In this case, the elements in each array are offset in the directionperpendicular to the scan direction by half the element dimension. Thisprovides overlap in the measurement images created by each sensingelement so that small flaws and defects cannot fall between the sensingelements when the sensor is scanned over the test material. FIG. 24shows a two-dimensional array 60 of sensing elements 62 placedthroughout the footprint of the drive winding segments 64. Note that thedrive winding segments 46 (FIG. 22), 54 (FIG. 23), and 64 (FIG. 24) ofeach respective embodiment are connected together with side connections,similar to the side connections 20 shown in FIG. 4, to maintain thecontinuity of current through the drive winding segments.

FIG. 25 shows a circular array 70 of sensing elements 72 placed aroundthe circumference of a drive winding 74 having rotational symmetry.Interconnections between each segment of the drive winding 74 are madewith conductor pairs similar to the conductor pairs 32 shown in FIG. 5.The locations of the sensing elements around the circumference of thecenter line 76 are staggered so that the effective scan paths for thesensing elements on the right and left sides of the sensor will overlapwhen the sensor is scanned over the test material. The sensor elementsmay be run with or without feedback loops and may be placed on aflexible support structure. The feedback secondary coils can take avariety of shapes, such as circular (FIG. 22) or square (FIG. 23). Forbiasing, a permanently magnetized flexible layer may be used. Inaddition, a magnetizable layer on the front or back of the sensor canalso be used to shape the magnetic field and/or improve sensitivity toproperty variations.

The position of the GMR elements within the feedback coil, and theposition of the feedback coil within the primary winding can also beadjusted. FIG. 26 illustrates that one or more GMR sensors 84 can besurrounded by a feedback coil 82 and placed at the center of a drivewinding 80. The use of multiple GMR sensors within the footprint of thedrive winding promotes imaging of material properties when the array isscanned in a direction perpendicular to the row of GMR sensors. The useof a single feedback coil and multiple GMR sensor elements eliminatescross-talk between elements, which may occur if each GMR element has itsown feedback coil, and also simplifies the drive circuitry for thesensor array. FIG. 27 shows a similar array with the row of GMR elements84 and feedback coil offset so that it is closer one side of the primarywinding than the other. This results in an asymmetric response when thearray is scanned over a flaw since the array is more sensitive to theeffects of the flaw when it passes beneath the nearer portion of theprimary winding. Similarly, sensing elements can be placed outside ofthe drive winding, as illustrated in FIG. 28, where the row of sensorelements 84 is far from the drive winding 80 while a second row ofsensors 86 is near the drive winding. An advantage of this configurationis that any connection leads to the sensing elements does not have topass over the conductors of the drive winding, which helps to minimizeparasitic responses. FIG. 29 shows a similar configuration with a shapedfield or distributed winding structure. The drive winding containsmultiple conductor segments 88 for imposing the magnetic field while oneor more GMR sensors 84 are placed within a feedback coil 82 within thedrive structure. The connection to a drive winding conductor 90 alsoshows that each of the drive winding conductor segments 88 areinterconnected. As with the single winding drives, the GMR sensors donot need to be placed at the center of the distributed drive structure.

The selection of the meandering or distributed single wavelength orhalf-wavelength (e.g., simple rectangle) winding design will varydepending on the depth of sensitivity required in a specific applicationand access issues for the sensor. For some of these designconsiderations, refer to U.S. Pat. Nos. 5,015,951, 5,453,689, and5,793,206. For example, detection of cracks under fastener heads shouldconsider the fastener head size, the spacing of the fasteners, and thelocation of the sensing elements and drives relative to the fasteners,to maximize sensitivity and minimize interferences. Also, the type ofdefect, such as corrosion versus fatigue or residual stress, will affectthe selection of the winding construct design.

In many situations, the sensor or sensor array will be in motionrelative to the test material. This can occur, for example, whenscanning across the surface of a material for a flaw. Most often thetime interval determined by the scanning speed and the characteristiclength scale of the sensor is much greater than the time period of theimposed AC field, in which case the effects of the motion arenegligible. However, in some situations the relative motion of thesensor and the material under test can influence the magnetic fielddistribution and the sensor response. For example, for a spatiallyperiodic winding distribution and material motion in the same direction,then the magnetic vector decays exponentially with distance into auniform material with a decay rate

γ=√{square root over (k ² +jσμ(ω−ku))}  (1)

where k=2π/λ is the wavenumber, σ is the electrical conductivity, μ isthe permeability, ω=2πf is the angular frequency of the excitation, andu is the material velocity. The velocity has the effect of changing theeffective frequency of the excitation and hence the decay rate of themagnetic field into the material under test. This effect can be modeledand FIG. 30 illustrates how the sensor transinductance, shown on amagnitude/phase plot, changes as the material velocity is increased. Oneinteresting result is that for a range of velocities the phase of thetransinductance becomes positive, which corresponds to an impedancehaving a negative real component. This is not unphysical and is theresult of the use of a single lumped element component representationfor the two-port representation of a continuum system. Similar resultshave been observed with dielectric sensors yielded a negativetransconductance. FIG. 31 shows a comparison in the magnetic vectorpotential between a moving and stationary metal layer. In addition tobreaking the symmetry with respect to the quarter-period point, themotion also acts to decrease the magnitude of the potential. While thesignificance of the motion depends upon the excitation frequency, thefundamental spatial wavelength of the sensor, and the velocity, Eq. 1suggests that low frequency measurements taken with high scan speeds (orvelocities) may be impacted by the relative motion.

The real-time estimate of material or geometric properties requires theuse of efficient table look-up and interpolation algorithms. Thesealgorithms convert the value of the measured transimpedance of a sensorto material properties (parameter estimation) by interpolating betweenthe points of a look-up table of pre-computed sensor responses (known asa measurement grid). The type of interpolation that needs to be carriedout for a grid look-up is not what is done in typical look-up tablealgorithms, which are forward interpolation algorithms. This is acritical distinction. Interpolation methods of this nature are discussedin (Press, 1992), though the subject of two-dimensional inverseinterpolation algorithms is not discussed.

The concept of inverse interpolation is best illustrated with anexample. Suppose there is a function y(x) whose values y_(n) are knownat a set of points x_(n). x is the independent variable and x_(n) arechosen by the designer of the look-up table to span the anticipatedfunction domain. The most common task is finding a value for y at agiven value of x. This is one-dimensional forward interpolation and isvery easy to carry out since it is known from the start which x_(n) isclosest to x.

Finding a value of x that corresponds to a given value of y in this sametable is known as inverse interpolation. It is a much more difficultoperation especially if y(x) is not a monotonic function. Furthermorethe values of y may be repeated and may not be evenly spaced.

In two dimensions, the forward interpolation entails finding the valueof a function y(x₁, x₂) given x₁ and x₂ by interpolating in atwo-dimensional table. This can be done via bilinear interpolation orits higher order variants. The key feature is that, as in theone-dimensional case, the closest point in the table and the table cellneeded for the interpolation is known from the onset. The actualinterpolation is also easy since the table is designed to form arectangular grid. Two-dimensional inverse interpolation comes about whenthere is a table that lists the values of two functions, y₁(x₁, x₂) andy₂(x₁, x₂) at a set of predetermined values of the independentvariables, forming a rectangular grid. The goal is to find theappropriate values for x₁ and x₂ where the functions y₁ and y₂ contain apair of known values. This inverse task is much more complicated thanthe forward task. Some of the complications are (1) even when both y₁and y₂ fall inside the ranges of the corresponding functions, there isno guarantee that a solution exists; (2) it is not known at the outsetwhat the indices of x₁ and x₂ are for the point closest to the target;and (3) even if the cell in the table that should be used forinterpolation is found, in general in y₁ and y₂ space this cell is notrectangular and no standard interpolation formula can be applied.

The starting point for the inverse interpolation is the identificationof the grid cell that contains the target point. One method foridentifying the closest grid point (in order to establish which cell touse in the interpolation) is to calculate the distance to each point onthe grid and pick the point with the smallest distance. This is veryinefficient because it requires calculating the distance to each pointin the grid for each table-up and can also give the wrong results forhighly elongated or non-orthogonal cell boundaries. A more efficientmethod is to judiciously choose a starting point and search in a localregion of the grid until the required grid cell is located. This can beimplemented in the following procedure:

-   1. Choose a starting point. Initiating the search close to the    target point can significantly improve the efficiency of the search.    In general, this should be the cell used in the last look-up,    because it is very likely that consecutive measurements lie close    together on the grid. In fact in most cases the last cell used    contains the current target point, so that this step often concludes    the search. If this is the first look-up, a rough interpolation is    done based on the four corners of the grid to estimate the indices    of the starting point, using method identical to the one described    below.-   2. Zoom out. If the target point is not inside the current cell,    increase the size of the area tested to 2×2, 4×4, etc. until either    the current grid “square” (collection of cells being considered)    contains the target point, or it reaches a predetermined limit or    the size of the entire grid.-   3. Move the grid square. If the grid square currently under    consideration does not contain the target point, move to the next    grid square of the current size in the direction toward the target    point. Keep moving until one of the following three conditions    occurs: (a) the target point falls inside the current square; (b)    the square reaches the edge of the grid; (c) it cannot be determined    which the correct direction to move is. This last condition can    occur in special cases near singular points, near grid folds where    the multiple grid cells contain the target point, or if the current    grid square is not a convex quadrangle.-   4. Zoom in. If the size of the current grid square is greater than a    single grid cell, reduce it by a factor of two and go back to    step 3. Otherwise, end the search.    If in the end the current grid cell contains the target point, the    search was successful. Otherwise, the target point is most likely    off the grid.

There can be cases when the search fails even if the target point is onthe grid. This can happen if the grid is so curved that moving in adirection toward the target in the lane of the grid (magnitude/phasespace) moves away from the correct cell in parameter space. These casesare handled by reverting to the “brute force” method of computing thedistance to every point on the grid and finding the closest grid pointto the target point. It is then used as the starting cell for a secondrun through the search algorithm. If in fact the target point is on thegrid, this will find the correct cell with a very high likelihood. Mostoften, however, the first run has failed because the point is off thegrid, so that if used every time a point is off the grid, this secondsearch step takes a disproportionately longtime to run, which can createproblems especially when performing real-time estimations. Whether togive up after the first try or attempt the long search method depends onthe particular application.

In order to determine the direction to move in step 3 above, a formula(Eq. 3) is applied. This calculates the area of the triangles formed byeach of the four sides of the grid square and the target point. The signof the result is used to determine whether the target point lies insidethe square, or else in which direction the square is to be moved. Forexample, if the target point is to the “right” of both the “left” and“right” sides of the square, it must be moved one over to the right,etc. The directions here are put in quotations, because the meanings of“right”, “left”, “top”, and “bottom” depend on the direction ofincreasing parameter index number, and on whether the grid is“right-handed” or “left-handed”, i.e. whether the axes inmagnitude/phase space and parameter space have the same or oppositesense.

After identifying the grid cell which contains the target point, thenext step is to interpolate within the grid to determine the parametervalues. A simple two-dimensional inverse interpolation is illustrated inFIG. 32. This interpolation method is based on the distances between thetarget point and the lines of constant parameter (P or Q):

$\begin{matrix}{{P = \frac{{d_{3}P_{1}} - {d_{1}P_{2}}}{d_{3} - d_{1}}}{Q = \frac{{d_{4}Q_{1}} - {d_{2}Q_{2}}}{d_{4} - d_{2}}}} & (2)\end{matrix}$

The distances d₁ through d₄ are calculated by dividing the area of thetriangle with vertices at the two corners and the target point by thelength of the cell side and then dividing by two. The following formulasis used to find the triangle area:

A _(ijk) =x ₁(y _(k) −y _(j))+x _(j)(y _(i) −y _(k))+x _(k)(y _(j) −y_(i))  (3)

The sign of A_(ijk) is positive or negative, depending on whether themotion is in a positive (counterclockwise) or negative (clockwise)direction along the circumscribed circle when going from i to j to k.Therefore this formula retains directionality information although itcalculates area. If all three points lie on a line, the result is zero.It is important to remember that the distances in Eq. 2 are signedquantities, so that when the target point lies inside the cell, d₁ andd₃ will have opposite signs, as do d₂ and d₄. The order of the areaindices in Eq. 3 is important. The advantage of maintaining the polarityinformation is that the correct result will be obtained even if thetarget point lies outside the cell used for the interpolation.

The problem with this “simple” inverse interpolation method iscontinuity across cell boundaries. The value of the parameter associatedwith the cell wall being crossed, e.g., P when crossing line 1-2 in FIG.32, will be continuous, because it is equal to the value of theparameter on that line (P₁). However, the value of the other parameter,e.g., Q when crossing line 1-2, will make a discontinuous jump as thegrid cell being used for interpolation changes. This is usually a smalljump since in most grids the walls of neighboring cells are nearlyparallel. The practical implications of this discontinuous behavior aremost pronounced when using iterative techniques to determine three ormore unknowns where it could lead to lack of convergence. It is alsopossible to observe the effect when scanning along a part withcontinuously varying properties, using a coarse grid where the measuredproperty may appear to change in step increments.

A more complex inverse interpolation algorithm can be used to solve thediscontinuity problem. This method relies on the fact that, given atarget point on a grid cell wall, both of the estimated parameters, Pand Q, must depend only on the coordinates of the target point and thetwo end points of the edge being crossed, common to both cells. Forexample, when the target point lies on line 1-2, both P and Q mustdepend only on the coordinates of points 1, 2, and T. With the simplemethod this is only true for P. This is accomplished using the methodillustrated in FIG. 33. Instead of using Eq 0.2 to calculate P and Qdirectly, it is used to find the coordinates of four new points, 5-8,one on each of the four sides of the cell using:

$\begin{matrix}{{x_{5} = \frac{{d_{4}x_{2}} - {d_{2}x_{1}}}{d_{4} - d_{2}}}{y_{5} = \frac{{d_{4}y_{2}} - {d_{2}y_{1}}}{d_{4} - d_{2}}}{x_{6} = \frac{{d_{3}x_{2}} - {d_{1}x_{3}}}{d_{3} - d_{1}}}{y_{6} = \frac{{d_{3}y_{2}} - {d_{1}y_{3}}}{d_{3} - d_{1}}}{x_{7} = \frac{{d_{4}x_{3}} - {d_{2}x_{4}}}{d_{4} - d_{2}}}{y_{7} = \frac{{d_{4}y_{3}} - {d_{2}y_{4}}}{d_{4} - d_{2}}}{x_{8} = \frac{{d_{3}x_{1}} - {d_{1}x_{4}}}{d_{3} - d_{1}}}{y_{8} = \frac{{d_{3}y_{1}} - {d_{1}y_{4}}}{d_{3} - d_{1}}}} & (4)\end{matrix}$

Now instead of being associated with a line segment, each of the twopairs of four grid parameters are associated with a pair of points, asshown in FIG. 34. If, for example, the target point lies on cell side1-2, points 8 and 6 will coincide with 1 and 2 respectively, making Qdependent only on these two points, as required. If P and Q areexpressed in terms of the coefficients a and b, using these equations:

P=(1−a)P ₁ +aP ₂ Q=(1−b)Q ₁ +bQ ₂  (5)

then standard bilinear interpolation leads to the following expressionsfor these coefficients:

$\begin{matrix}{{a = \frac{{\left( {x_{8} - x_{6}} \right)\left( {y - y_{5}} \right)} - {\left( {x - x_{5}} \right)\left( {y_{8} - y_{6}} \right)}}{{\left( {x_{8} - x_{6}} \right)\left( {y_{7} - y_{5}} \right)} - {\left( {x_{7} - x_{5}} \right)\left( {y_{8} - y_{6}} \right)}}}{b = \frac{{\left( {x - x_{6}} \right)\left( {y_{7} - y_{5}} \right)} - {\left( {x_{7} - x_{5}} \right)\left( {y - y_{6}} \right)}}{{\left( {x_{8} - x_{6}} \right)\left( {y_{7} - y_{5}} \right)} - {\left( {x_{7} - x_{5}} \right)\left( {y_{8} - y_{6}} \right)}}}} & (6)\end{matrix}$

Using this procedure, the grid cell has effectively been transformedfrom an arbitrary quadrangle to a parallelepiped (outlined with dashedlines in FIG. 34), although the shape of this parallelogram depends onthe coordinates of the target point.

The simple method also could not accommodate cases when two adjacentcorners of a grid cell coincided, as shown in FIG. 35. This can be aserious impediment since grids with triangular cells appear to be quitecommon. Often, this type of measurement grid arises when the “air” point(zero conductivity for magnetic grids or a relative permittivity of onefor dielectric grids) is included in the grid, so that the sensorresponse does not change with “lift-off” and the entire edge of the gridfolds into a single point. This air point can be used as a reference forsensor calibration, necessitating its inclusion in the grids to avoidthe need for special treatment.

As a working example consider the case where points 3 and 4 coincide, asshown in FIG. 35. The main problem with applying the method outlined inthe previous section directly is that the distance d₃ is not welldefined, because points 3 and 4 do not uniquely define a line. It is notpossible simply to use the distance between the target point T and point3, because the polarity information is lost in this case (among otherthings) and the results would be wrong, especially if the target isoutside the cell, which we hope to handle with at least some grace. Thesolution is to take the direction of the line through 3 and 4 to beparallel to the line through 1 and 2. A consequence of this is that theline through 6 and 8 will always be parallel to 1-2 and will passthrough the target point T. The undefined distances need to be redefinedvia the segments as shown in FIG. 35.

Another special consideration is the cases when a grid folds in onitself This look-up algorithm attempts to handle this kind of grid, atleast in the simplest cases when there is no more than one “fold” or“twist”. The main difficulty is that the searching stage of thealgorithm may become confused near the fold line. For example, at thefold many grid cells are quadrangles whose opposite edges cross. It isnot possible to determine whether the target point is “left” or “right”of a cell whose left and right edges intersect. In fact, it is notpossible to decide whether the target point falls inside the cell atall. The approach taken in such cases is to try four different runs ofthe search algorithm described above, starting at the four corners ofthe grid. It has been found that this approach is quite effective athandling most cases, providing even for the possibility of finding twosolutions in the areas where the grid surface overlaps itself. Noattempt is made to determine which one of the two solutions is to beconsidered the “correct” one, as such decisions need to be made at ahigher level, using other information.

Fast computation methods also facilitate the real-time estimate ofmaterial or geometric properties. These methods formulate the physicaland mathematical models that separate the computations of intermediateresults that depend on sensor geometry from those that depend upon theproperties of the material under test. This allows the calculation ofmeasurement grids or the running of parameter estimation routines basedon minimization algorithms to be much more efficient. More specifically,in a typical calculation of the sensor response, intermediatecalculations based solely on sensor geometry only need to be calculatedonce and not many times. These results can be stored and recalled asnecessary, which can be much faster than the actual computation of theintermediate results and can provide dramatic decreases in processingtimes for iterative calculations. Iterative calculations are performedover a range of material or geometric properties during measurement gridgeneration and when minimizing the error between measurement andsimulation data in parameter estimation routines.

The inventions described here relate to methods and apparatus for thenondestructive measurements of materials using sensors that applyelectromagnetic fields to a test material and detect changes in theelectromagnetic fields due to the proximity and properties of the testmaterial. Although the discussion focused on magnetoquasistatic sensors,many of the concepts extend directly to electroquasistatic sensors aswell.

While the inventions has been particularly shown and described withreference to preferred embodiments thereof, it will be understood tothose skilled in the art that various changes in form and details may bemade therein without departing from the spirit and scope of theinvention as defined by the appended claims.

References incorporated herein by reference in their entirety:

-   Goldfine, N. G. (1990) “Uncalibrated, Absolute Property Estimation    and Measurement Optimization for Conducting and Magnetic Media Using    Imposed w-k Magnetometry,” Ph. D. thesis, Department of Mechanical    Engineering, Massachusetts Institute of Technology.-   Goldfine, N. G. (1993) “Magnetometers for Improved Characterization    in Aerospace Applications,” Materials Evaluation, 51 (3).-   Hood, R., L. M. Falicov, (1993) “Theory of the negative    magnetoresistance in magnetic metallic multilayers,” MRS Symposium    Proceedings vol. 313.-   Press, W. H., B. P. Flannery, S. A. Teukolsky, W. T.    Vetterling (1992) “Numerical Recipes; The Art of Scientific    Computing.”-   Rempt, R., (2001) presented at the 2001 Aeromat Conference, Jun.    11-14, 2001.-   Wincheski, B., J. Simpson, M. Namkung, D. Perey, E. Scales, and R.    Louie (2001) presented at the 2001 QNDE Conference, New Brunswick,    Me.

The following documents are also incorporated herein by reference intheir entirety.

-   1. NASA Phase I Proposal Titled “Shaped Field Giant Magnetoresistive    Sensor Arrays for Material Testing”, Topic #A1.05-8767, dated Jun.    5, 2001.-   2. Technical Paper titled “Flexible Eddy Current Sensors and    Scanning Arrays for Inspection of Steel and Alloy Components”,    presented at the 7^(th) EPRI Steam Turbine/Generator Workshop and    Vendor Exposition, Aug. 20-23, 2001.-   3. Technical Paper titled “High-Resolution Eddy Current Sensor    Arrays for Detection of Hidden Damage including Corrosion and    Fatigue Cracks”, presented at the NASA/FAA/DoD Conference on Aging    Aircraft, Sep. 10-13, 2001.-   4. Presentation Slides titled “High-Resolution Eddy Current Sensor    Arrays with Inductive and Magnetoresistive Sensing Elements”,    presented at the ASNT Fall Conference, Oct. 15-19, 2001.-   5. Massachusetts Institute of Technology Doctoral Thesis (2001),    titled “Deep Penetration Magnetoquasistatic Sensors,” by Yanko    Sheiretov.

1. A test circuit for measuring material properties comprising: aprimary winding configured to carry an electric current; a solid-statesensing element for sensing a property of a magnetic field produced whenthe primary winding is driven by the electric current; a feedback coilthat encircles the solid-state sensing element; and a feedback controlcircuit configured to control a bias current provided to the feedbackcoil.
 2. The test circuit of claim 1, wherein the solid-state sensingelement is a magnetoresistive sensing element.
 3. The test circuit ofclaim 2, wherein the solid-state sensing element is a giantmagnetoresistive sensing element.
 4. The test circuit of claim 1,wherein the solid-state sensing element is a hall effect sensor.
 5. Thetest circuit of claim 1, wherein a segment of the primary winding mostproximal to the solid-state sensing element is linear.
 6. The testcircuit of claim 1, wherein the feedback control circuit is configuredto control the bias current such that an output of the solid-statesensing element has a linear response to the property of the magneticfield.
 7. The test circuit of claim 1, wherein the property of themagnetic field is the amplitude of the magnetic field.
 8. The testcircuit of claim 1, wherein the solid-state sensing element is one of aplurality of solid-state sensing elements forming a array.
 9. A methodof characterizing a test material with test circuit of claim 1, themethod comprising: placing the solid-state sensing element near the testmaterial; driving the primary winding with the electric current;providing a bias current to the feedback coil; measuring an output ofthe solid-state sensing element; and estimating a property of the testmaterial from the output of the solid-state sensing element.
 10. A testcircuit comprising: a primary winding configured to carry an electriccurrent; an array of solid-state sensing elements for sensing a propertyof a magnetic field produced when the primary winding is driven by theelectric current; a feedback coil that encircles at least one of thesolid-state sensing elements; and a control circuit configured tocontrol a bias current provided to the feedback coil.
 11. The testcircuit of claim 10 wherein the primary winding has a linear segment andthe array is aligned with, but offset from, the linear segment of theprimary winding.
 12. The test circuit of claim 10 wherein the primarywinding has a circular segment and the solid-state elements aredistributed about a circle having a same center as the circular segmentof the primary winding.
 13. The test circuit of claim 10, wherein thearray further comprises at least one inductive coil sensing element. 14.The test circuit of claim 10, wherein the primary winding has a linearsegment and a distance between the linear segment and the solid-statesensing elements varies.
 15. The test circuit of claim 10, wherein eachof the solid-state sensing elements is encircled by a respectivefeedback coil, each of the respective feedback coils having its owncontrol circuit.
 16. The test circuit of claim 10, wherein the feedbackcoil encircles at least two solid-state sensing elements.
 17. The testcircuit of claim 10, wherein the solid-state sensing elements compriseat least one magnetoresistive sensing element.
 18. The test circuit ofclaim 10, wherein the solid-state sensing elements comprise at least onegiant magnetoresistive sensing element.
 19. The test circuit of claim18, wherein the array comprises: a first giant magnetoresistive sensingelement having a first orientation so as to measure the magnetic fieldin a direction; and a second giant magnetoresistive sensing elementhaving a second orientation, different from the first, so as to measurethe magnetic field in a direction.
 20. The test circuit of claim 10,wherein the solid-state sensing elements comprise at least one halleffect sensor.